一类分层三角剖分下三次样条空间的维数

本文定义了平面单连通多边形域的一类较任意的三角剖分－分层三角剖分，并通过分析二元样条的积分协调条件，确定了分层三角剖分卜三次Ｃ^１作条函数空间的维数．     

A remark on interpolation by bivariate splines

We study the determining set for bivariate spline spacesS _k ^o on type-1 triangulation of square using B-net techniques. We further construct the interpolation schemes for these spline spaces that are unisolvent for any function f of C^σ.     

Geometric interpolation by planar cubic <Emphasis Type="Italic">G</Emphasis><Superscript>1</Superscript> splines

In this paper, geometric interpolation by G ¹ cubic spline is studied. A wide class of sufficient conditions that admit a G ¹ cubic spline interpolant is determined. In particular, convex data as well as data with inflection points are included. The existence requirements are based upon geometric properties of data entirely, and can be easily verified in advance. The algorithm that carries out the verification is added. AMS subject classification (2000)  65D05, 65D07, 65D17     

Bivariate spline method for numerical solution of steady state Navier–Stokes equations over polygons in stream function formulation

We use the bivariate spline finite elements to numerically solve the steady state Navier–Stokes equations. The bivariate spline finite element space we use in this article is the space of splines of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the steady state Navier–Stokes equations is employed. Galerkin's method is applied to the resulting nonlinear fourth‐order equation, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in H²(Ω) of the nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The Galerkin method with C¹ cubic splines is implemented in MATLAB. Our numerical experiments show that the method is effective and efficient. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 147–183, 2000     

W-加密三角剖分下二元五次超样条函数空间的局部Lagrange插值

本文选取二元五次C2超样条函数空间作为插值空间,考虑局部Lagrange插值.首先对三角剖分△进行着色,通过Wang-加密三角剖分对原剖分△细分大约一半的三角形.然后通过在内边增加一些另外的光滑条件,使得样条函数在某些边上达到更高阶的光滑.最后在△的加密三角剖分内选择Lagrange插值点.结果表明相应的插值基函数具有...     

Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

In this paper we generate and study new cubature formulas based on spline quasi-interpolants defined as linear combinations of C ¹ bivariate quadratic B-splines on a rectangular domain Ω, endowed with a non-uniform criss-cross triangulation, with discrete linear functionals as coefficients. Such B-splines have their supports contained in Ω and there is no data point outside this domain. Numerical results illustrate the methods.     

On structure of bivariate spline space of lower degree over arbitrary triangulation

The structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property. A new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces. The upper bound of the dimension of spline spaces over triangulation given by L.L. Schumaker is slightly improved via the new vertex coding method. The structure of multivariate spline spaces and over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by Luo and Wang. A kind of sufficient conditions on judging non-singularity of the and spaces over arbitrary triangulation is given, which only depends on the topological property of the triangulation. From the sufficient conditions, a triangulation strategy is presented at the end of the paper. The strategy ensures that the constructed triangulation is non-singular (or generic) for and .     

Bivariate spline method for numerical solution of time evolution Navier‐Stokes equations over polygons in stream function formulation

We use a bivariate spline method to solve the time evolution Navier‐Stokes equations numerically. The bivariate splines we use in this article are in the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier‐Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth‐order equation, Crank‐Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L₂(0, T; H²(Ω)) ∩ L_∞(0, T; H¹(Ω)) of the 2D nonlinear fourth‐order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C¹ cubic splines are implemented in MATLAB for solving the Navier‐Stokes equations numerically. Our numerical experiments show that the method is effective and efficient. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 776–827, 2003.     

Morgan-Scott型剖分上的样条空间的奇异性

Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.     

二元样条的积分表示及分层三角剖分下二次样条空间的维数

本文通过引入一个积分协调条件，首次给出了二元样条的一个积分表示．文中还定义了平面单连通多边形区域的所谓分层三角剖分，并确定了此剖分下二次样条空间的维数．     

THE DIMENSION OF A CLASS OF BIVARIATE SPLINE SPACES

ＴＨＥＤＩＭＥＮＳＩＯＮＯＦＡＣＬＡＳＳＯＦＢＩＶＡＲＩＡＴＥＳＰＬＩＮＥＳＰＡＣＥＳ￥ＧＡＯＪＵＮＢＩＮＡｂｓｔｒａｃｔ：ＷｅｅｓｔａｂｌｉｓｈｔｈｅｄｉｍｅｎｓｉｏｎｆｏｒｍｕｌａｏｆｔｈｅｓｐａｃｅｏｆＣｒｂｉｖａｒｉａｔｅｐｉｅｃｅｗｉｓｅｐ...     

On the construction of local quadratic spline quasi-interpolants on bounded rectangular domains

In this paper local bivariate C¹$C^1$ spline quasi-interpolants on a criss-cross triangulation of bounded rectangular domains are considered and a computational procedure for their construction is proposed. Numerical and graphical tests are provided.     

C2-rational cubic spline involving tension parameters

In the present paper, C¹-piecewise rational cubic spline function involving tension parameters is considered which produces a monotonie interpolant to a given monotonie data set. It is observed that under certain conditions the interpolant preserves the convexity property of the data set. The existence and uniqueness of a C²-rational cubic spline interpolant are established. The error analysis of the spline interpolant is also given.     

Range restricted interpolation using Gregory's rational cubic splines

The construction of range restricted univariate and bivariate interpolants to gridded data is considered. We apply Gregory's rational cubic C¹ splines as well as related rational quintic C² splines. Assume that the lower and upper obstacles are compatible with the data set. Then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range restricted interpolation is successful.     

On explicit basis of bivariate spline space

For a subdivision Δ of a region in d-dimensional Euclidean space, we consider computation of dimension and of basis function in spline space S _k ^r (Δ) consisting of all C piecewise polynomial functions over Δ of degree at most k. A computational scheme is presented for computing the dimension and bases of spline space S _k ^r (Δ). This scheme based on the Grobner basis algorithm and the smooth co-factor method for computing multivariate spline. For bivariate splines, explicit basis functions of S _k ^r (Δ) are obtained for any integer k and r when Δ is a cross-cut partition. The Project is partly supported by the Science and Technology New Star Plan of Beijing and Education Committee of Beijing.     

研究二元样条空间维数的Blossoming方法

本文综述了研究二元样条的Ｂｌｏｓｓｏｍｉｎｇ方法．成功地重建了平面上贯穿剖分的维数公式．而且利用这种方法，对定义在Ｍｏｒｇａｎ－Ｓｃｏｔ剖分上样条空间的维数取得了一些新的结果．     

关于二元样条空间下的三角剖分的分解

本文指出，在一定条件下，对于一个二元样条空间，所考虑的三种剖分中的某些胞腔和网线可以消去，而前后两个三角剖分下样条空间的结构有着紧密的联系，从而可以用简单划分下的空间结构表示复杂剖分下的空间结构。该分解剖分的步骤可以递推的进行，尤其对Ｓ＾１ｓ。据此，本文还分析了剖分对Ｓ＾１２的奇异性并给出一组奇异的剖分。     

The instability degree in the diemnsion of spaces of bivariate spline

In this paper, the dimension of the spaces of bivariate spline with degree less that 2r and smoothness order r on the Morgan-Scott triangulation is considered. The concept of the instability degree in the dimension of spaces of bivariate spline is presented. The results in the paper make us conjecture the instability degree in the dimension of spaces of bivariate spline is infinity.     

Variational Interpolation of Subsets

We consider the problem of the variational interpolation of subsets of Euclidean spaces by curves such that the L² norm of the second derivative is minimized. It is well-known that the resulting curves are cubic spline curves. We study geometric boundary conditions arising for various types of subsets such as subspaces, polyhedra, and submanifolds, and we indicate how solutions can be computed in the case of convex polyhedra.     

A normalized representation of super splines of arbitrary degree on Powell–Sabin triangulations

In the paper, a family of bivariate super spline spaces of arbitrary degree defined on a triangulation with Powell–Sabin refinement is introduced. It includes known spaces of arbitrary smoothness r and degree \(3r-1\) but provides also other choices of spline degree for the same r which, in particular, generalize a known space of \(\mathscr {C}^{1}\) cubic super splines. Minimal determining sets of the proposed super spline spaces of arbitrary degree are presented, and the interpolation problems that uniquely specify their elements are provided. Furthermore, a normalized representation of the discussed splines is considered. It is based on the definition of basis functions that have local supports, are nonnegative, and form a partition of unity. The basis functions share numerous similarities with classical univariate B-splines.